What relevance do imaginary numbers have
to the real world?

Okay, now we've seen that imaginary numbers exist. However, they exist in
the context of a different number system, something different from the number
systems we are used to. The "complex numbers" that make up this system
are pairs of numbers;
do they really deserve to be called "numbers" in their
own right?

Well, remember that fractions are pairs of numbers also. They clearly
deserve to be called numbers in their own right, since they can
measure "how much" in some contexts (for instance, "I ate three
quarters of a pie"). So, the principle of considering a pair of
numbers (in this example, 3 and 4) as a number in its own right is
well established.

The fact remains, though, that complex numbers have much less
direct relevance to real-world quantities than other numbers
do. An imaginary number could not be used as a measurement of how much
water is in a bottle, or how far a kite has travelled, or how many
fingers one has.

Nonetheless, there are a few real world quantities for which complex
numbers are the natural model. The strength of an electromagnetic
field is one example. The field has both an electric and a magnetic
component, so it takes a pair of real numbers (one for the intensity
of the electric field, one for the intensity of the magnetic field) to
describe the field strength. This pair of real numbers can be thought
of as a complex number, and it turns out that the strange rule of
multiplication of complex numbers has relevance to the physics of an
electromagnetic field.

Although such direct applications of complex numbers to the real world
are few, their indirect applications are many. Many properties
related to real numbers only become clear when the real numbers are
thought of as sitting inside the complex number system. Therefore,
complex numbers aid in the understanding even of things that are
described by ordinary, familiar real numbers.

It's like trying to understand a shadow. The shadow lives in a
two-dimensional world, so only two-dimensional concepts are directly
applicable to it. However, thinking of the three-dimensional object
casting the shadow can aid in understanding it, even though
three-dimensional concepts don't have any direct application to the
two-dimensional world of the shadow. Likewise, complex numbers may not
be directly applicable to a real world measurement any more than a
three-dimensional object is directly applicable to a 2-dimensional
shadow, but they can still help us understand it.

Here's another analogy that may be helpful. Consider the following
argument:

Population A has 236 people, 48 of them children.

Population B has 1234 people, 123 of them children.

The fraction of children in population A is 48/236 while the
fraction of children in population B is 123/1234.

48/236 (approx. 0.2) is much greater than 123/1234 (approx. 0.1).

Therefore, population A is, on the whole, a much younger population
than population B.

This argument has used fractions, non-integer numbers, in a
problem where they have no physical relevance! You can't measure
populations in fractions; you can't have "half a person", for example.
The kind of numbers that have direct relevance to measuring numbers of
people are the natural numbers; fractions are just as alien to this
context as the complex numbers are alien to most real-world
measurements. And yet, despite this, allowing ourselves to move from
the natural numbers to the larger set of rational numbers enabled us
to deduce something about the real world situation, even though
measurements in that particular real world situation only involve
natural numbers.

In the same way, being willing to think about what happens in the
larger set of complex numbers allows us to draw conclusions about real
world situations even when actual measurements in that particular real
world situation only involve the real numbers. This happens all the
time in engineering applications. If you've seem some calculus, you
may be able to understand the following example: in engineering and
other applications, one very often has to solve an equation of the
form a y" + b y' + c y = 0 (*) for the unknown function y.

It turns out that there's a way to get the solutions provided one can
solve the quadratic equation a r^2 + b r + c = 0 for the variable
r. In the real numbers, there may not be any solutions. However, in
the complex numbers there are, so one can find all complex-valued
solutions to the equation (*), and then finally restrict oneself to
those that are purely real-valued. The starting and ending points of
the argument involve only real numbers, but one can't get from the
start to the end without going through the complex numbers. Since
equations like (*) need to be solved all the time in real-life
applications such as engineering, complex numbers are needed.